Finite Group Databasebeta

← #192748

Subgroups — #192748

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8895C214yesmaximal, complemented
7#8883C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15#2355—22no—
16#2355—22no—
17#2355—22no—
18#2355C2214yes—
19C2×#391—21no—
20C2×#387—22no—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948C2212yes—
29#1947—22no—
30#1947C2212yes—
31#1852—22no—
32#1845C2212yes—
33#1834C2212yes—
34#1824—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1487C418yescomplemented
41C2×Q8⋊C8C2214yes—
42#1296C2214yes—
43#1279C2214yes—
44C23×Q8D411yescharacteristic
45C22×M16—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C414yes—
51C2×C4×Q8D412yes—
52C2×C4×Q8Q814yes—
53C2×C4×Q8C2312yes—
54C2×#117—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2×C414yes—
70#428—44no—
71C2×#107—22no—
72C2×#107—24no—
73C2×#106—22no—
74#423—22no—
75#423C2×C414yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2312yes—
86#391—24no—
87#391—24no—
88#387—28no—
89#387—28no—
90#384—21no—
91#384—21no—
92#384D412yes—
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#384C2312yes—
96#382—24no—
97#382—24no—
98#382—21no—
99#380—21no—
100#380—21no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—22no—
119#340—22no—
120Q8⋊C8C2×C4116yescomplemented
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C4⋊C414yes—
126C22×Q8C2×D412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—24no—
136C2×M16—42no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8C4⋊C418yes—
149C4×Q8C22×C414yes—
150C4×Q8C2×Q814yes—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C4⋊C414yes—
180C2×C4⋊C4Q1614yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—41no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—28no—
193C2×C42—44no—
194C2×C42—41no—
195C2×C42—84no—
196C2×C42C22×C412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204#107—44no—
205#107—84no—
206#106—44no—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8C22×C414yes—
213#103—44no—
214#103—44no—
215C22⋊C8—24no—
216C22⋊C8—42no—
217C4×C8—22no—
218C4×C8—42no—
219C4×C8C22×C414yes—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—22no—
224#95—22no—
225#95—41no—
226#95—41no—
227C24#12011yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—48no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—48no—
238C2×Q8C2×C4⋊C414yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×C4⋊C412yes—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4C22⋊Q812yes—
272C22×C4C22⋊Q812yes—
273C22×C4#12312yes—
274C22×C4C2×Q1612yes—
275C22×C4#13512yes—
276C22×C4AGL(1,ℤ/8)12yes—
277C22×C4#13712yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281M16—48no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4C2×C4⋊C414yes—
317C4⋊C4C2×Q1614yes—
318C22⋊C4—82no—
319C42—24no—
320C42—24no—
321C42—28no—
322C42—44no—
323C42—44no—
324C42—88no—
325C42—82no—
326C42C2×C4⋊C412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42#12312yes—
332C42C23×C411yescharacteristic
333C42C22×Q811yescharacteristic
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#38512yes—
341C23#44912yes—
342C23#45112yes—
343C23C2×#12011yescharacteristic
344Q8—24no—
345Q8—24no—
346Q8—48no—
347Q8—42no—
348Q8—42no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#38512yes—
386C2×C4#38512yes—
387C2×C4#38512yes—
388C2×C4C4×Q1614yes—
389C2×C4#44014yes—
390C2×C4#44912yes—
391C2×C4#45112yes—
392C2×C4#47714yes—
393C2×C4#47914yes—
394C2×C4C22×C4⋊C411yescharacteristic
395C2×C4C2×C4×D411yescharacteristic
396C2×C4C2×C4×D411yescharacteristic
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4C2×#12311yescharacteristic
400C2×C4C22×Q1611yescharacteristic
401C2×C4C2×#13511yescharacteristic
402C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57512yes—
405C2×C4#57512yes—
406C8—44no—
407C8—44no—
408C8—82no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22#186314yes—
416C22C2×#38511yescharacteristic
417C22C2×#45111yescharacteristic
418C22C2×#44911yescharacteristic
419C22#300511yescharacteristic
420C22#300511yescharacteristic
421C22#301911yescharacteristic
422C22#302011yescharacteristic
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—44no—
428C4—44no—
429C4—44no—
430C4—84no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4#186314yes—
435C4#291112yes—
436C4C2×C4×Q1611yescharacteristic
437C4C2×#44011yescharacteristic
438C4C2×#47711yescharacteristic
439C4C2×#47911yescharacteristic
440C4#309312yes—
441C2—24no—
442C2#1874911yescharacteristic
443C2#1875011yescharacteristic
444C2#1877911yescharacteristic
445C2#1878011yescharacteristic
446C2#2084511yescharacteristic
447C2#2086411yescharacteristic
448C2C2×#186311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #8895C2semidirect
7. #8883C2semidirect
40. #1487C4semidirect
120. Q8⋊C8C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23552
#219822
#211702
#207342
C2×#14872
#88952
#88832
#80432
#78922